The bill passed by a mostly bipartisan vote of 220 to 210.
The legislation, numbered "H.R. 1" for the importance Democrats attach to it, "is designed to restore the voices of Americans who felt left out and locked out for too long," its original sponsor, Representative John Sarbanes, said in remarks outside the U.S. Capitol before the vote.
The bill is one of many the House Democrats are voting on early in the Congress on a number of priorities, such as lesbian, gay, bisexual and transgender rights, policing and the environment.
President Joe Biden, a Democrat, has said he would sign the bill into law if it cleared both the House and the Senate.
But the bills face long odds in the Senate, where all 48 Democrats and the two independents who caucus with them would need to be joined by 10 Republican senators to overcome a filibuster.
"We're going to do everything in our power - I think Chuck Schumer's going to do everything in his power - to make sure that the agenda that we are working on and promised the people will in fact be the agenda we put forward and hopefully pass," Representative Steny Hoyer, the No. 2 Democrat, told a news conference on Tuesday.
Already, some Democrats have trained fire on the filibuster and called for its elimination. Hoyer called it "undemocratic" on Tuesday, while James Clyburn, the No. 3 Democrat, described how it had been used to deny rights to Black citizens.
A move to destroy the filibuster would face severe opposition. Democratic Senator Joe Manchin has repeatedly said he would "never" vote to get rid of it, while Biden has said he opposes overturning it.
Some Democrats have suggested a compromise of a carve-out could be possible.
Asked about the possibility of a carve-out for voting rights issues, Georgia Senator Raphael Warnock told a reporter on Tuesday, "I think that the issues are urgent enough to leave all options on the table."
Democrats have argued the legislation is necessary to lower barriers to voting and make the US political system more democratic and responsive to voters.
But Republicans say it would take powers away from the states and fail to do enough to combat fraud, and the influential right-wing Heritage Foundation think tank has urged lawmakers to vote against it.
The bill has particular importance this year because the U.S. Census Bureau is set to turn over population data states use to redraw congressional district maps ahead of the 2022 elections, in which every House lawmaker's seat is up for grabs.
States use different criteria and procedures for drawing the maps. In 33 states, state legislatures control at least part of the process, the National Conference of State Legislatures says.
Parties in control of state legislatures have often wielded the power to draw the maps in a way that benefits the party in power. Some states have also used this process to target Black voters.
The measure also comes as lawmakers in 43 states have introduced legislation to place more limits on voting, according to the Brennan Center for Justice.
Several states, including Georgia, where Warnock and Democrat Jon Ossoff were elected in a political upset in January, have since introduced legislation to change election procedures that activists say make it harder for people to vote.